多數值類神經元與週期性活化函數(Multi-valued Neuron with Periodic activation function，MVN-P)在這裡也簡稱為週期性多數值類神經元，MVN-P是由Aizenberg所提出用以解決分類問題的一種神經元架構。在MVN-P之中，每兩種不同類別區塊之間的邊界都是固定的，這樣子的因素可能會使得整個神經元在訓練的過程中產生難以收斂或者是沒有辦法收斂的情形。在本篇論文中我們提出了兩個修改MVN-P的模型，而這兩種模型都是基於設計”邊界沒那麼固定”的點子所設計的。我們所提出的第一種模型中，在每兩種不同類別之間的邊界加入了稱為Crisp Buffer的容忍區域，在訓練階段，只要有分類非正確的資料位於此區就可以被容忍其錯誤。在我們所提出的第二種模型中，我們使用了Fuzzy Buffer來做為判斷是否容錯的機制，在訓練階段分類錯誤的資料其歸屬函數(Membership degree)若低於所設定之Threshold，則可以被容忍錯誤。接著我們使用Genetic Algorithm來最佳化上述兩種模型之參數，減少使用者自訂參數的負擔。除此之外，MVN-P難以用以處理資料類別數量較多的分類問題，因此我們發展了一個樹狀架構來克服上述的情況，最後實驗的結果可以得知我們所提出的方法是有效的。

Multi-valued Neuron with Periodic activation function (MVN-P) was proposed by Aizenberg for solving classification problems. The boundaries between two distinct categories are crisply specified in MVN-P, which may result in slow convergence or being unable to converge at all in the learning process. In this paper, we propose two revised models of MVN-P based on the idea of un-sharp boundaries. In the first revised model, a crisp buffer is provided around a boundary between two distinct categories, allowing incorrect assignments in the buffer to be tolerated in the training phase. In the second revised model, a fuzzy buffer is provided instead and an incorrect assignment with membership degree less than a Threshold can be tolerated. Genetic algorithms are applied to derive optimal values for the parameters involved in different models, alleviating the burden of setting them manually by the user. Besides, MVN-P has difficulties solving the classification problems having a large number of categories. A tree structure is developed to overcome these difficulties. Simulations have been done and the results are presented to demonstrate the effectiveness of our proposed ideas.

論文審定書 i
致謝 iii
摘要 iv
Abstract v
圖次 viii
表次 x
第一章 導論 1
1.1 研究動機與文獻探討 1
1.2 論文架構 4
第二章 文獻回顧 5
2.1 多數值類神經元 5
2.2 週期性多數值類神經元 9
第三章 週期性多數值類神經元與容忍區域 14
3.1 MVN-P with Crisp Buffer 14
3.2 MVN-P with Fuzzy Buffer 23
3.3 基因演算法與多數值類神經元之最佳化 27
3.4 實驗方法 33
3.5 實驗結果 – l與效能的關係 35
3.6 實驗結果 – 神經元與GA 39
第四章 多類別問題 41
4.1 1-a-a 41
4.2 樹狀結構 43
4.3 實驗結果 46
第五章 多數值類神經元活化函數之變形 48
5.1 MVN-Sin 49
5.2 實驗結果 51
第六章 結論與未來研究方向 53
參考文獻 54

[1] Sun, E.-P. Lim, W.-K. Ng, and J. Srivastava, ``Blocking reduction strategies in hierarchical text classification,' IEEE Transactions on Knowledge and Data Engineering, vol. 16, no. 10, pp. 1305--1308, 2004.
[2] L. Rokach and O. Maimon, ``Top-down induction of decision trees classifiers-A survey,' IEEE Transactions on Systems, Man, and Cybernetics, Part C： Applications and Reviews, vol. 35, no. 4, pp. 476--487, 2005.
[3] T. Windeatt, R. Duangsoithong and R. Smith, ``Embedded feature ranking for ensemble MLP classifiers,' IEEE Transactions on Neural Networks, vol. 22, no. 6, pp. 988-994, 2011.
[4] H. Guo and S. B. Gelfand, ``Classification trees with neural network feature extraction,' IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 923--933, 1992.
[5] A. Hart, ``Using neural networks for classification tasks -- some experiments on datasets and practical advice,' The Journal of the Operational Research Society, vol. 43, no. 3, pp. 215--226, 1992.
[6] B. Cheng and D. M. Titterington, ``Neural networks： A review from a statistical perspective,' Statistical Science, vol. 9, no. 1, pp. 2--30, 1994.
[7] B. D. Ripley, ``Neural networks and related methods for classification,' Journal of the Royal Statistical Society. Series B (Methodological), vol. 56, no. 3, pp. 409--456, 1994.
[8] R. Anand, K. Mehrotra, C. K. Mohan, and S. Ranka, ``Efficient classification for multiclass problems using modular neural networks,' IEEE Transactions on Neural Networks, vol. 6, no. 1, pp. 117--124, 1995.
[9] Y. Pei, Neural Network： A Powerful Tool for Classification. Germany： World Scientific, 1995.
[10] C.-Y. Cheng, K.-H. Lin, and C.-W. Shih, ``Multistability in recurrent neural networks,' SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1301--1320, 2006.
[11] Z.-H. Zhou and X.-Y. Liu, ``Training cost-sensitive neural networks with methods addressing the class imbalance problem,' IEEE Transactions on Knowledge and Data Engineering, vol. 18, no. 1, pp. 63--77, 2006.
[12] S. Raudys, R. Kybartas, and E. K. Zavadskas, ``Multicategory nets of single-layer perceptrons： Complexity and sample-size issues,' IEEE Transactions on Neural Networks, vol. 21, no. 5, pp. 784--795, 2010.
[13] A. K. Jain and J. Mao, ``A k-nearest neighbor artificial neural network classifier,' in Proc. Int. Joint Conf. Neural Networks, Seattle, WA, 1991, pp. 515--520.
[14] Y. Bao, X. Du, and N. Ishii, ``Improving performance of the k-nearest neighbor classifier by tolerant rough sets,' in Proc. Int. Symp. Cooperative Database Systems for Advanced Applications, Beijing, 2001, pp. 167--171.
[15] D. Kolbe, Q. Zhu, and S. Pramanik, ``On k-nearest neighbor searching in non-ordered discrete data spaces,' in Proc. Int. Conf. Data Engineering, I stanbul, 2007, pp. 426--435.
[16] M. Z. Al-Faiz, A. A. Ali, and A. H. Miry, ``A k-nearest neighbor based algorithm for human arm movements recognition using EMG signals,' in Proc. Int. Conf. Energy, Power and Control, Basrah, 2010, pp. 159--167.
[17] N. N. Aizenberg and I. N. Aizenberg, ``CNN based on multi-valued neuron as a model of associative memory for grey-scale images,' in Proc. Int. Workshop Cellular Neural Networks, Munich, 1992, pp. 36--41.
[18] I. Aizenberg, D. V. Paliy, J. M. Zurada and J. T. Astola, ``Blur identification by multilayer neural network based on multivalued neurons,' IEEE Transactions on Neural Networks, vol. 19, no. 5, pp. 883--898, 2008.
[19] I. Aizenberg, ``Periodic activation function and a modified learning algorithm for the multivalued neuron,' IEEE Transactions on Neural Networks, vol. 21, no. 12, pp. 1939--1949, 2010.
[20] I. Aizenberg, Complex-Valued Neural Networks with Multi-Valued Neurons. USA： Springer, 2011.
[21] M. L. Wong, Y. Yam, and P. Baranyi, ``Representing membership functions as elements in function space,' in Proc. American Control Conf., Arlington, 2001, vol. 3, pp. 1922--1927.
[22] Y. Yam, M. L. Wong, and P. Baranyi, ``Interpolation with function space representation of membership functions,' IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 398--411, 2006.
[23] L. A. Zadeh, ``Fuzzy sets,' Information and Control, vol. 8, no. 3, pp. 338--353, 2010.
[24] H. Ying, ``Deriving analytical input--output relationship for fuzzy controllers using arbitrary input fuzzy sets and zadeh fuzzy and operator,' IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 654--662, 2006.
[25] M. Pota, M. Esposito, and G. De Pietro, ``Transformation of probability distribution into fuzzy set interpretable with likelihood view,' in Proc. Int. Conf. Hybrid Intelligent Systems, Melacca, 2011, pp. 91--96.
[26] T. F. Eibert and V. Hansen, ``Triangular and rectangular elements in the spectral-domain analysis of arbitrarily shaped planar circuits,' IEEE Transactions on Antennas and Propagation, vol. 41, no. 8, pp. 1145--1147, 1993.
[27] N. Afshordi and M. R. Meybodi, ``Using learning automata for tuning fuzzy membership functions in learning driver preferences,' in Proc. Int. Conf. Intelligent and Advanced Systems, Kuala Lumpur, 2007, pp. 87--92.
[28] A. M. Aziz, ``Effects of fuzzy membership function shapes on clustering performance in multisensor-multitarget data fusion systems,' in Proc. IEEE Int. Conf. Fuzzy Systems, Jeju Island, 2009, pp. 1839--1844.
[29] A. M. Murshid and S. A. Loan, ``Architectural design of fuzzy inference processor using triangular-shaped membership function,' in Proc. IEEE Conf. Open Systems, Langkawi, 2011, pp. 16--20.
[30] J. H. Holland, Adaptation in Natural and Artificial System. Cambridge, MA： MIT Press, 1992.
[31] N. Chaiyaratana and A. M. S. Zalzala, ``Recent developments in evolutionary and genetic algorithms： Theory and applications,' in Proc. IEEE Int. Conf. Genetic Algorithms in Engineering System： Innovations and Applications, Glasgow, 1997, pp. 270--277.
[32] S. Abedi and R. Tafazolli, ``Genetically modified multiuser detection for code division multiple access systems,' IEEE Journal on Selected Areas in Communications, vol. 20, no. 2, pp. 463-473, 2002.
[33] G. F. Luger, Artificial Intelligence. USA： Addison Wesley, 2002.
[34] C. Y. Hou C. Y. R. Chen and U. Singh, ``Optimal algorithms for bubble sort based non-manhattan channel routing,' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 13, no. 5, pp. 603--609, 1994.
[35] G. Lin and X. Yao, ``Analysing crossover operators by search step size,' in Proc. IEEE Int. Conf. Evolutionary Computation, Indianapolis, 1997, pp. 107--110.
[36] C. Pitangui and G. Zaverucha, ``Improved natural crossover operators in GBIVIL,' in Proc. IEEE Int. Conf. Evolutionary Computation, Singapore, 2007, pp. 2157--2164.
[37] D. E. van de Vlag and A. Stein, ``Incorporating uncertainty via hierarchical classification using fuzzy decision trees,' IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 1, pp. 237--245, 2007.
[38] K. E. Graves and R. Nagarajah, ``Uncertainty estimation using fuzzy measures for multiclass classification,' IEEE Transactions on Neural Networks,vol. 18, no. 1, pp. 128--140, 2007.
[39] K. Ito and K. Xiong, Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control, vol. 45, no. 5, pp. 910--927, 2000.
[40] G. Heo and P. Gader, ``Learning the number of Gaussian components using hypothesis test,' in Proc. Int. Joint Conf. Neural Networks, Atlanta, 2009, pp. 1206--1212.
[41] J. He, H. Gu, and S. Jiang, ``Twin gaussian processes for binary classification,' in Proc. IEEE Int. Conf. Data Mining, Vancouver, 2011, pp. 1074--1079.
[42] P. G. Espejo, S. Ventura and F. Herrera, ``A survey on the application of genetic programming to classification,' IEEE Transactions on Systems, Man, and Cybernetics, Part C： Applications and Reviews, vol. 40, no. 2, pp. 121--144, 2010.
[43] R. C. Barros, M. P. Basgalupp, A. C. P. L. F. de Carvalho, and A. A. Freitas, ``A survey of evolutionary algorithms for decision-tree induction,' IEEE Transactions on Systems, Man, and Cybernetics, Part C： Applications and Reviews, vol. 42, no. 3, pp. 291--312, 2012.
[44] C. Scott and R. D. Nowak, ``Minimax-optimal classification with dyadic decision trees,' IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1335--1353, 2006.
[45] H. Lim, H. N. Chu, and C. Yim, ``Hierarchical binary search tree for packet classification,' IEEE Communications Letters, vol. 11, no. 8, pp. 689--691, 2007.
[46] K. S. Balagani and V. V. Phoha, ``On the relationship between dependence tree classification error and Bayes error rate,' IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 29, no. 10, pp. 1866--1868, 2007.
[47] B, Liu, Z. Hao, and E. C. C. Tsang, ``Nesting one-against-one algorithm based on SVMs for pattern classification,' IEEE Transactions on Neural Networks, vol. 19, no. 12, pp. 2044--2052, 2008.
[48] S. Hashemi, Y. Yang, Z. Mirzamomen, and M. Kangavari, ``Adapted one-versus-all decision trees for data stream classification,' IEEE Transactions on Knowledge and Data Engineering, vol. 21, no. 5, pp. 624--637, 2009.
[49] C.-N. Young, C.-W. Yen, Y.-H. Pao, and M. L. Nagurka, ``One-class-at-a-time removal sequence planning method for multiclass classification problems,' IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1544--1549, 2006.
[50] UCI Machine Learning Repository,
http：//archive.ics.uci.edu/ml/index.html.
[51] Y. Yu and Y. Xinjie, ``Cooperative Coevolutionary Genetic Algorithm for Digital IIR Filter Design,' IEEE Transactions on Industrial Electronics, vol. 54, no. 3, pp. 1311—1318, 2007.
[52] J. A. Vasconcelos, J. A. Ramirez, R. H. C. Takahashi and R. R. Saldanha, ``Improvements in Genetic Algorithms,' IEEE Transactions on Magnetics, vol. 37, no. 5, pp. 3414—3417, 2001.